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August 6, 2009 Author, Jon Kleinberg, was recently cited in the New York Times for his statistical analysis research in the Internet age. Algorithm Design introduces algorithms by looking at the real-world problems that motivate them. The book teaches students a range of design and analysis techniques for problems that arise in computing applications. The text encourages an understanding of the algorithm design process and an appreciation of the role of algorithms in the broader field of computer science. Features + Benefits Focus on problem analysis and design techniques. Discussion is grounded in concrete problems and examples rather than abstract presentation of principles, with representative problems woven throughout the text. Over 200 well crafted problems from companies such as Yahoo!® and Oracle®. Each problem has been class tested for usefulness and accuracy in the authors' own undergraduate algorithms courses. Broad coverage of algorithms for dealing with NP-hard problems and the application of randomization, increasingly important topics in algorithms. Algorithm DesignJon Kleinberg and Eva TardosTable of Contents 1 Introduction: Some Representative Problems 1.1 A First Problem: Stable Matching 1.2 Five Representative Problems Solved ExercisesExcercisesNotes and Further Reading2 Basics of Algorithms Analysis 2.1 Computational Tractability 2.2 Asymptotic Order of Growth Notation 2.3 Implementing the Stable Matching Algorithm using Lists and Arrays 2.4 A Survey of Common Running Times 2.5 A More Complex Data Structure: Priority Queues Solved Exercises Exercises Notes and Further Reading3 Graphs 3.1 Basic Definitions and Applications 3.2 Graph Connectivity and Graph Traversal 3.3 Implementing Graph Traversal using Queues and Stacks 3.4 Testing Bipartiteness: An Application of Breadth-First Search 3.5 Connectivity in Directed Graphs 3.6 Directed Acyclic Graphs and Topological Ordering Solved Exercises Exercises Notes and Further Reading 4 Divide and Conquer 4.1 A First Recurrence: The Mergesort Algorithm 4.2 Further Recurrence Relations 4.3 Counting Inversions 4.4 Finding the Closest Pair of Points 4.5 Integer Multiplication 4.6 Convolutions and The Fast Fourier Transform Solved Exercises Exercises Notes and Further Reading5 Greedy Algorithms 5.1 Interval Scheduling: The Greedy Algorithm Stays Ahead 5.2 Scheduling to Minimize Lateness: An Exchange Argument 5.3 Optimal Caching: A More Complex Exchange Argument 5.4 Shortest Paths in a Graph 5.5 The Minimum Spanning Tree Problem 5.6 Implementing Kruskal's Algorithm: The Union-Find Data Structure 5.7 Clustering 5.8 Huffman Codes and the Problem of Data Compression*5.9 Minimum-Cost Arborescences: A Multi-Phase Greedy Algorithm Solved Exercises Excercises Notes and Further Reading 6 Dynamic Programming 6.1 Weighted Interval Scheduling: A Recursive Procedure 6.2 Weighted Interval Scheduling: Iterating over Sub-Problems 6.3 Segmented Least Squares: Multi-way Choices 6.4 Subset Sums and Knapsacks: Adding a Variable 6.5 RNA Secondary Structure: Dynamic Programming Over Intervals 6.6 Sequence Alignment 6.7 Sequence Alignment in Linear Space 6.8 Shortest Paths in a Graph 6.9 Shortest Paths and Distance Vector Protocols *6.10 Negative Cycles in a Graph Solved ExercisesExercisesNotes and Further Reading7 Network Flow 7.1 The Maximum Flow Problem and the Ford-Fulkerson Algorithm 7.2 Maximum Flows and Minimum Cuts in a Network 7.3 Choosing Good Augmenting Paths *7.4 The Preflow-Push Maximum Flow Algorithm 7.5 A First Application: The Bipartite Matching Problem 7.6 Disjoint Paths in Directed and Undirected Graphs 7.7 Extensions to the Maximum Flow Problem 7.8 Survey Design 7.9 Airline Scheduling 7.10 Image Segmentation 7.11 Project Selection 7.12 Baseball Elimination *7.13 A Further Direction: Adding Costs to the Matching Problem Solved ExercisesExercisesNotes and Further Reading 8 NP and Computational Intractability 8.1 Polynomial-Time Reductions 8.2 Reductions via "Gadgets": The Satisfiability Problem 8.3 Efficient Certification and the Definition of NP 8.4 NP-Complete Problems 8.5 Sequencing Problems 8.6 Partitioning Problems 8.7 Graph Coloring8.8 Numerical Problems 8.9 Co-NP and the Asymmetry of NP8.10 A Partial Taxonomy of Hard Problems Solved Exercises Exercises Notes and Further Reading9 PSPACE: A Class of Problems Beyond NP9.1 PSPACE 9.2 Some Hard Problems in PSPACE 9.3 Solving Quantified Problems and Games in Polynomial Space9.4 Solving the Planning Problem in Polynomial Space9.5 Proving Problems PSPACE-Complete Solved ExercisesExercisesNotes and Further Reading 10 Extending the Limits of Tractability 10.1 Finding Small Vertex Covers 10.2 Solving NP-Hard Problem on Trees 10.3 Coloring a Set of Circular Arcs *10.4 Tree Decompositions of Graphs *10.5 Constructing a Tree Decomposition Solved Exercises Exercises Notes and Further Reading11 Approximation Algorithms 11.1 Greedy Algorithms and Bounds on the Optimum: A Load Balancing Problem 11.2 The Center Selection Problem 11.3 Set Cover: A General Greedy Heuristic 11.4 The Pricing Method: Vertex Cover 11.5 Maximization via the Pricing method: The Disjoint Paths Problem 11.6 Linear Programming and Rounding: An Application to Vertex Cover *11.7 Load Balancing Revisited: A More Advanced LP Application 11.8 Arbitrarily Good Approximations: the Knapsack Problem Solved ExercisesExercisesNotes and Further Reading 12 Randomized Algorithms 12.1 A First Application: Contention Resolution 12.2 Finding the Global Minimum Cut 12.3 Random Variables and their Expectations 12.4 A Randomized Approximation Algorithm for MAX 3-SAT 12.5 Randomized Divide-and-Conquer: Median-Finding and Quicksort 12.6 Hashing: A Randomized Implementation of Dictionaries 12.7 Finding the Closest Pair of Points: A Randomized Approach 12.8 Randomized Caching 12.9 Chernoff Bounds 12.10 Load Balancing *12.11 Packet Routing 12.12 Background: Some Basic Probability DefinitionsSolved ExercisesExercisesNotes and Further Reading 13 Local Search 13.1 The Landscape of an Optimization Problem 13.2 The Metropolis Algorithm and Simulated Annealing 13.3 An Application of Local Search to Hopfield Neural Networks 13.4 Maximum Cut Approximation via Local Search 13.5 Choosing a Neighbor Relation *13.6 Classification via Local Search 13.7 Best-Response Dynamics and Nash EquilibriaSolved ExercisesExercisesNotes aAugust 6, 2009 Author, Jon Kleinberg, was recently cited in the for his statistical analysis research in the Internet age. Algorithm Design introduces algorithms by looking at the real-world problems that motivate them. The book teaches students a range of design and analysis techniques for problems that arise in computing applications. The text encourages an understanding of the algorithm design process and an appreciation of the role of algorithms in the broader field of computer science.

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This book studies two optimization problems, maximum satisfiability and planing of satisfiability. The maximum satisfiability problem (max-SAT) is the optimization counterpart of the satisfiability problem (SAT). The goal of max-SAT is to maximize the number of clauses satisfied. planning as satisfiability is a class of planning aiming to achieve a plan with optimal resource, cost, or makespan by using the SAT approach. We present a mix- SAT formulation for these two optimization problems and examine to extend the Davis-Putnam-Logemann- Loveland (DPLL) procedure, which is the basic framework for the original SAT problem, for this mix- SAT formulation. We progressively develop a series of algorithms and reconsider many general SAT techniques for these two optimization problems.

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This thesis deals with a general modeling framework for large-scale biological systems which is, on the one hand, applied to various practical instances, and on the other hand, strictly formalized and mathematically analyzed with respect to its complexity and structure. For the biological application initially an overview of existing analytic methods for biological systems is presented, and the proposed modeling framework is classified in this context. The framework is based on logical implication formulas. It allows for the verification of a biological model, the prediction of its response to prescribed stimuli, as well as the identification of possible intervention strategies for diseases or failure modes. This basic model is afterwards extended into two directions: First, timing information of reactions in the biological unit are incorporated. This generalization additionally enables to detect possible unknown timing information or inconsistencies that arise due to modeling errors. Besides this, it provides a method to consistently integrate the logical models of related biological units into one model. Second, the purely binary basic framework is enhanced by including a fine discretization of a biological component's activity level. This permits to express different effects depending on different levels of activity of one component, and therefore the predictions of the model become more sophisticated. On the mathematical side the logical framework and its extensions are derived and formalized. The basic model evolves to a special type of satisfiability problem (SAT) whose complexity is classified to be generally hard but mathematically easy subclasses are identified. The correspondence between SAT and integer programming is exploited and the underlying polyhedra are analyzed. Interestingly, the SAT problem allows for a wider class of polynomially solvable problems than its integer programming equivalent. Nevertheless, the computational results provided proof that the integer programming approach is computationally feasible. The basic SAT problem can additionally be translated into a bipartite digraph for which algorithms are adapted, and their practical use is discussed. Furthermore, for a special class of biological units a duality framework based on linear programming duality is derived, which completes the theory of such biological units. The dynamic extension of the basic framework yields a related SAT problem that contains the original one as a special case, and is thus hard to solve as well. The focus for this extension is on the analysis of maximally feasible and minimally infeasible solutions of the extended SAT problem. Therefore, it is necessary to optimize over the set of solutions of the SAT problem which suggests to employ the equivalent integer programming approach. To enumerate all maximally feasible and minimally infeasible solutions the Joint Generation algorithm is utilized. To this end, a monotone reformulation of the extended SAT problem is derived that preserves the maximally feasible and minimally infeasible solutions, and at the same time significantly reduces the size of the description. In certain very restrictive cases the resulting integer optimization problems are even computationally tractable. Finally, the minimally infeasible solutions are completely characterized by means of graph structures in the original digraph, and an alternative method for computing all minimally infeasible solutions via polyhedral projection is obtained. The discrete extension of the logical framework leads to a generalization of the SAT problem, the so called interval satisfiability problem. In this setting the variables are integer valued and associated intervals provide the set of values for which the expression becomes TRUE. To computationally determine feasible solutions, this problem is transformed to a system of polynomials which can be checked for feasibility by means of Hilbert's Nullstellensatz. Moreover, the general interval satisfiability problem is analyzed with respect to complexity and satisfiability. Concerning the computational complexity, it is shown to be generally hard even if assuming certain restrictions for the formulas. Concerning the satisfiability behavior the well known threshold phenomenon of classical random SAT, which has been observed for interval satisfiability, is examined and lower bounds on specific thresholds are identified.

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A decision procedure is an algorithm that, given a decision problem, terminates with a correct yes/no answer. Here, the authors focus on theories that are expressive enough to model real problems, but are still decidable. Specifically, the book concentrates on decision procedures for first-order theories that are commonly used in automated verification and reasoning, theorem-proving, compiler optimization and operations research. The techniques described in the book draw from fields such as graph theory and logic, and are routinely used in industry. The authors introduce the basic terminology of satisfiability modulo theories and then, in separate chapters, study decision procedures for each of the following theories: propositional logic, equalities and uninterpreted functions, linear arithmetic, bit vectors, arrays, pointer logic, and quantified formulas. They also study the problem of deciding combined theories and dedicate a chapter to modern techniques based on an interplay between a SAT solver and a decision procedure for the investigated theory.This textbook has been used to teach undergraduate and graduate courses at ETH Zurich, at the Technion, Haifa, and at the University of Oxford. Each chapter includes a detailed bibliography and exercises. Lecturers' slides and a C++ library for rapid prototyping of decision procedures are available from the authors' website.

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This book constitutes the refereed proceedings of the 14th Conference on Computability in Europe, CiE 2018, held in Kiel, Germany, in July/ August 2017.The 26 revised full papers were carefully reviewed and selected from 55 submissions. In addition, this volume includes 15 invited papers. The conference CiE 2018 has six special sessions, namely: Approximation and optimization, Bioinformatics and bio-inspired computing, computing with imperfect information, continuous computation, history and philosophy of computing (celebrating the 80 th birthday of Martin Davis), and SAT-solving.

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Although they are believed to be unsolvable in general, tractability results suggest that some practical NP-hard problems can be efficiently solved. Combinatorial search algorithms are designed to efficiently explore the usually large solution space of these instances by reducing the search space to feasible regions and using heuristics to efficiently explore these regions. Various mathematical formalisms may be used to express and tackle combinatorial problems, among them the constraint satisfaction problem (CSP) and the propositional satisfiability problem (SAT). These algorithms, or constraint solvers, apply search space reduction through inference techniques, use activity-based heuristics to guide exploration, diversify the searches through frequent restarts, and often learn from their mistakes.In this book the author focuses on knowledge sharing in combinatorial search, the capacity to generate and exploit meaningful information, such as redundant constraints, heuristic hints, and performance measures, during search, which can dramatically improve the performance of a constraint solver. Information can be shared between multiple constraint solvers simultaneously working on the same instance, or information can help achieve good performance while solving a large set of related instances. In the first case, information sharing has to be performed at the expense of the underlying search effort, since a solver has to stop its main effort to prepare and communicate the information to other solvers, on the other hand, not sharing information can incur a cost for the whole system, with solvers potentially exploring unfeasible spaces discovered by other solvers. In the second case, sharing performance measures can be done with little overhead, and the goal is to be able to tune a constraint solver in relation to the characteristics of a new instance - this corresponds to the selection of the most suitable algorithm for solving a given instance.The book is suitable for researchers, practitioners, and graduate students working in the areas of optimization, search, constraints, and computational complexity.

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This book constitutes the refereed proceedings of the 17th European Conference on Evolutionary Computation in Combinatorial Optimization, EvoCOP 2017, held in Amsterdam, The Netherlands, in April 2017, co-located with the Evo 2017 events EuroGP, EvoMUSART and EvoApplications.The 16 revised full papers presented were carefully reviewed and selected from 39 submissions. The papers cover both empirical and theoretical studies on a wide range of academic and real-world applications. The methods include evolutionary and memetic algorithms, large neighborhood search, estimation of distribution algorithms, beam search, ant colony optimization, hyper-heuristics and matheuristics. Applications include both traditional domains, such as knapsack problem, vehicle routing, scheduling problems and SAT, and newer domains such as the traveling thief problem, location planning for car-sharing systems and spacecraft trajectory optimization. Papers also study important concepts such as pseudo-backbones, phase transitions in local optima networks, and the analysis of operators. This wide range of topics makes the EvoCOP proceedings an important source for current research trends in combinatorial optimization.

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VLSI CADhas greatly bene?ted from the use of reduced ordered Binary Decision Diagrams (BDDs) and the clausal representation as a problem of Boolean Satis?ability (SAT), e.g. in logic synthesis, ver- cation or design-for-testability. In recent practical applications, BDDs are optimized with respect to new objective functions for design space exploration. The latest trends show a growing number of proposals to fuse the concepts of BDD and SAT. This book gives a modern presentation of the established as well as of recent concepts. Latest results in BDD optimization are given, c- ering di?erent aspects of paths in BDDs and the use of e?cient lower bounds during optimization. The presented algorithms include Branch ? and Bound and the generic A -algorithm as e?cient techniques to - plore large search spaces. ? The A -algorithm originates from Arti?cial Intelligence (AI), and the EDA community has been unaware of this concept for a long time. Re- ? cently, the A -algorithm has been introduced as a new paradigm to explore design spaces in VLSI CAD. Besides AI search techniques, the book also discusses the relation to another ?eld of activity bordered to VLSI CAD and BDD optimization: the clausal representation as a SAT problem.

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